D-branes and Closed String Field Theory

N. Ishibashi (University of Tsukuba)

In collaboration with Y. Baba(Tsukuba) and K. Murakami(KEK)

JHEP 05(2006):029 JHEP 05(2007):020 JHEP 10(2007):008

理研シンポジウム「弦の場の理論07」 Oct. 7, 2007 1 §1 Introduction

D-branes in string field theory

• D-branes can be realized as soliton solutions in open string field theory

• D-branes in closed string field theory?

Hashimoto and Hata

HIKKO

♦A BRS invariant source term ♦tension of the brane cannot be fixed 2 Tensions of D-branes

=

Can one fix the normalization of the boundary states without using open strings?

For noncritical strings, the answer is yes. Fukuma and Yahikozawa

Let us describe their construction using SFT 3 for noncritical strings. c=0 noncritical strings

+ + double scaling limit

String Field l

Describing the matrix model in terms of this field, we obtain the string field theory for c=0 noncritical strings Kawai, N.I. 4 Stochastic quantization of the matrix model

Jevicki and Rodrigues

joining-splitting interactions 5 correlation functions

Virasoro constraints FKN, DVV

Virasoro constraints Schwinger-Dyson equations for the correlation functions

~ various vacua

6 Solitonic operators Fukuma and Yahikozawa Hanada, Hayakawa, Kawai, Kuroki, Matsuo, Tada and N.I.

These coefficients are chosen so that

From one solution to the Virasoro constraint, one can generate another by the solitonic operator. 7 These solitonic operators correspond to D-branes

amplitudes with ZZ-branes

state with D(-1)-brane = ghost D-brane 8 Okuda,Takayanagi critical strings?

noncritical case is simple

idempotency equation Kishimoto, Matsuo, Watanabe

For boundary states, things may not be so complicated 9 If we have ・SFT with length variable ・nonlinear equation like the Virasoro constraint for critical strings, similar construction will be possible.

We will show that ・ for OSp invariant SFT for the critical bosonic strings ・ we can construct BRS invariant observables using the boundary states for D-branes imitating the construction of the solitonic operators for the noncritical string case.

♦the BRS invariant observables →BRS invariant source terms~D-branes ♦the tensions of the branes are fixed

10 Plan of the talk

§2 OSp Invariant String Field Theory

§3 Observables and Correlation Functions

§4 D-brane States

§5 Disk Amplitudes

§6 Conclusion and Discussion

11 §2 OSp Invariant String Field Theory

light-cone gauge SFT

O(25,1) symmetry

OSp invariant SFT =light-cone SFT with Grassmann Siegel, Uehara, Neveu, West, Zwiebach, Kugo, Kawano,…. OSp(27,1|2) symmetry

12 variables

13 action

14 15 OSp theory 〜 covariant string theory with extra time and length

HIKKO

・no ・extra variables

16 The “action” cannot be considered as the usual action

looks like the action for stochastic quantization

BRS symmetry

the string field Hamiltonian is BRS exact 17 18 OSp invariant SFT 〜stochastic formulation of string theory ?

Green’s functions of BRS invariant observables ∥

Green’s functions in 26D ▪Wick rotation

▪1 particle pole

S-matrix elements in 26D ∥ ? S-matrix elements derived from the light-cone gauge SFT

19 §3 Observables and Correlation Functions

observables

ignoring the multi-string contribution

if

we need the cohomology of 20 cohomology of

(on-shell)

This state corresponds to a particle with mass 21 Observables

22 Free propagator

two point function

We would like to show that the lowest order contribution to this two point function coincides with the free propagator of the particle corresponding to this state.

23 light-cone quantization

24 25 the Hamiltonian is BRS exact a representative of the class

26 N point functions

we would like to look for the singularity

light-cone diagram

27 the singularity comes from

28 Repeating this for we get

29 Rem. higher order corrections

・two point function

・N point function

can also be treated in the same way at least formally 30 Wick rotation

・LHS is an analytic function of

・Wick rotation + OSp(27,1|2) trans.

・

reproduces the light-cone gauge result 31 §3 D-brane States Let us construct off-shell BRS invariant states, imitating the construction of the solitonic operator in the noncritical case.

Boundary states in OSp theory

boundary state for a flat Dp-brane

32 BRS invariant regularization

states with one soliton

will be fixed by requiring

33 creation

34 shorthand notation

35 Using these, we obtain

We need the idempotency equation.

36 Idempotency equations

leading order in

corrections 37 Derivation of the idempotency equations

38 39 ・ : quadratic in can be evaluated from the Neumann function

・ can also be evaluated from the Neumann function

・

40 we can fix the normalization in the limit

can be evaluated in the same way

41 Using these relations we obtain

if

Later we will show 42 States with N solitons

Imposing the BRS invariance, we obtain

looks like the matrix model 43 can be identified with the open string tachyon

may be identified with the eigenvalues of the matrix valued tachyon

open string tachyon

44 More generally we obtain

・we can also construct

45 §4 Disk Amplitudes

BRS invariant source term

Let us calculate amplitudes in the presence of

such a source term. 46 saddle point approximation

generate boundaries on the worldsheet

47 correlation functions

= ++…

+ +…

Let us calculate the disk amplitude 48 Rem.

In our first paper, we calculated the vacuum amplitude

▪we implicitly introduced two D-branes by taking the bra and ket

→ states with even number of D-branes

but ▪vacuum amplitude is problematic in light-cone type formulations

▪we overlooked a factor of 2 49 disk two point function

50 tachyon-tachyon

coincides with the disk amplitude up to normalization

・more general disk amplitudes can be calculated in the same way

51 low-energy effective action

Comparing this with the action for D-branes

52 §5 Conclusion and Discussion

♦D-brane BRS invariant source term

♦other amplitudes

♦similar construction for superstrings

we need to construct OSp SFT for superstrings

♦more fundamental formulation?

53